Remark 1.13 Borel's original characterization of base \(b\)  normality is (iv) (see [1]}). The equivalence of (i) and (iv) was shown by Niven and Zuckerman ([2]}).
The characterizations (v) and (vi) were established, respectively, by Pillai ([3]}) and Long ([4]}). For more details, see Chapter 1, Section 8 in [5]} and Appendix in [6]}.

